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The ring of integers OK is a finitely-generated Z-module. Indeed it is a freeZ-module, and thus has an integral basis, that is a basisb1, … ,bn ∈ OK of the Q-vector space K such that each element x in OK can be uniquely represented as

with ai ∈ Z.[3] The rank n of OK as a free Z-module is equal to the degree of K over Q.

In a ring of integers, every element has a factorisation into irreducible elements, but the ring need not have the property of unique factorisation: for example, in the ring of integers ℤ[√-5] the element 6 has two essentially different factorisations into irreducibles:[7][4]

One defines the ring of integers of a non-archimedean local field F as the set of all elements of F with absolute value ≤1; this is a ring because of the strong triangle inequality.[10] If F is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers. The ring of integers of an algebraic number field may be characterised as the elements which are integers in every non-archimedean completion.[2]

For example, the p-adic integers Zp are the ring of integers of the p-adic numbersQp.

^The ring of integers, without specifying the field, refers to the ring Z of "ordinary" integers, the prototypical object for all those rings. It is a consequence of the ambiguity of the word "integer" in abstract algebra.